mechanical vibrations explained

vibration isolation base excitation

When the base shakes, the response ratio is the transmissibility TR=X/Y; isolation (TR<1) needs r>√2, so lower ωₙ (soft mount) — and damping is double-edged

Resonance in A4 pushed the mass directly, but in real life it is the floor that shakes. Road bumps, an earthquake, a machine turning next door. When the base moves by Y and the mass moves by X, the ratio, the transmissibility TR = XY, is the heart of it. Isolation, the shaking actually getting smaller, begins only after the driving frequency passes √2 times the natural frequency. Before that boundary the floor's motion grows instead.

Instead of pushing the mass, shake the floor. Change the drive ratio r = ωωₙ and three things appear. When it is slow (r≪1) the mass follows the floor as it is (TR≈1). Near resonance (r=1) it shakes even more (TR≫1). When it is fast (r large) the mass stays almost still (TR<1, isolation). The faster the floor, the more the mass stays where it was.

The transmissibility curve is the map. Move r and a resonance peak rises at r=1. At r=√2 the curve passes exactly through TR=1 no matter the damping. Only to the right of that, where r>√2, does a real isolation region with TR<1 open. The part drawn in green is where protection begins.

How do you isolate, then? The driving frequency is already fixed, so you soften the mount, stiffness k↓ → ωₙ↓, and push r past √2. While the stiffness falls you pass through resonance (r=1) once and the motion swells, but go softer still and the mass grows quiet. The secret of isolation is a soft support.

Damping is a double-edged tool. Raise the damping ratio and the resonance peak falls, but inside the isolation region (r>√2) TR rises instead and the isolation gets worse. Every curve meeting at the one point r=√2 and TR=1 is the evidence. A mount therefore takes only enough damping to survive resonance at startup and shutdown, and avoids any more.

The same transmissibility governs both directions. Step through the examples and you meet a car suspension (road→body), an engine or washing-machine mount (machine→floor, and here it is force that is transmitted), and a bench for a precision instrument or a microscope (floor→equipment). Whether you protect the mass from the floor or the floor from the machine, the curve is the same and the prescription is the same. Lower the natural frequency ωₙ.

In PracticeVibration isolation is a cousin of resonance (A4). The difference is that the floor shakes, not a force. The heart is the transmissibility TR = XY = √[(1+(2ζr)²)((1−r²)²+(2ζr)²)], with r = ωωₙ. If r≪1 then TR≈1 and the mass follows the floor. If r=1 there is resonance. If r=√2 then TR=1 regardless of damping. Only if r>√2 is TR<1 and isolation real. So the rule is to lower the natural frequency enough, a soft support that sends the driving frequency above √2·ωₙ. Damping is double-edged. It lowers the resonance peak but makes the isolation region worse, so you give only enough to pass startup and shutdown. Displacement transmissibility = force transmissibility, so protecting equipment from a shaking floor and keeping a machine's vibration off the floor are the same problem. Suspensions, engine mounts, washing machines, seismic isolation, and microscope benches all sit on one curve.

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